
and \(G(r) = D^2(r) - H^2(r)\) . The parameters \(\omega \)  and \(m\)  are such that \(\omega ^2 > 0\)  and \(-\infty \le m^2 \le +\infty \) . The Gödel-type metrics are defined by the two parameters \(\omega \)  and \(m\) . Note that, identical pairs \((\omega , m)\)  specify isometric space-times [1]}. The standard Gödel solution is a particular case of the \(m^2>0\)  class of space-times in which \(m^2=2\omega ^2\) . However, causal and non-causal regions are allowed. These regions are determined from free parameters of the metric \(m\)  and \(\omega \) , and limited by a critical radius \(r_c\) . From this critical radius, there is a violation of causality. It is defined as
\(\sinh ^2 \left(\frac{mr_c}{2}\right) \ = \ \left(\frac{4\omega ^2}{m^2} -1 \right)^{-1}.\) 
